Skip to main content
SpringerLink
Log in

First integrals for Finsler metrics with vanishing \(\chi \)-curvature

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We prove that in a Finsler manifold with vanishing \(\chi \)-curvature (in particular with constant flag curvature) some non-Riemannian geometric structures are geodesically invariant and hence they induce a set of non-Riemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald curvature, or as functions of the mean Cartan torsion and the mean Landsberg curvature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Álvarez-Paiva, J.C.: Rigidity results for geodesically reversible Finsler metrics, arXiv:2106.10095

  2. Akbar-Zadeh, H.: Sur les espaces de Finsler à corbures sectionelles constantes. Acad. Roy. Belg. Bull. Sci. 74(5), 281–322 (1988)

    MATH  Google Scholar 

  3. Berwald, L.: On Finsler and Cartan geometries. III. Two-dimensional Finsler spaces with rectilinear extremals. Ann. Math. 42(1), 84–112 (1941)

    Article  MathSciNet  Google Scholar 

  4. Bucataru, I.: Invariant volume forms and first integrals for geodesically equivalent Finsler metrics. Proc. Amer. Math. Soc., https://doi.org/10.1090/proc/15961

  5. Bucataru, I., Constantinescu, O.: Generalized Helmholtz conditions for non-conservative Lagrangian systems. Math. Phys. Anal. Geom. 18(1), 1–24 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bucataru, I., Creţu, G.: A general version of Beltrami’s theorem in Finslerian setting. Publ. Math. Debrecen 97(3–4), 439–447 (2020)

    Article  MathSciNet  Google Scholar 

  7. Bucataru, I., Constantinescu, O.A., Creţu, G.: A class of Finsler metrics admitting first integrals. J. Geom Phys. 166, 104254 (2021)

    Article  MathSciNet  Google Scholar 

  8. Bucataru, I., Dahl, M.: Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. J. Geom. Mech. 1(2), 159–180 (2009)

    Article  MathSciNet  Google Scholar 

  9. Deicke, A.: Über die Finsler-Rüme mit \(A_i=0\). Arch. Math. 4, 45–51 (1953)

    Article  MathSciNet  Google Scholar 

  10. Foulon, P., Ruggiero, R.O.: A first integral for \(C^{\infty }\), \(k\)-basic Finsler surfaces and applications to rigidity. Proc. Amer. Math. Soc. 144(9), 3847–3858 (2016)

    Article  MathSciNet  Google Scholar 

  11. Grifone, J.: Structure presque-tangente et connexions I. Ann. Inst. Fourier 22, 287–334 (1972)

    Article  MathSciNet  Google Scholar 

  12. Grifone, J., Muzsnay, Z.: Variational Principles for Second-Order Differential Equations. Application of the Spencer theory to characterize variational sprays. World Scientific, River Edge, NJ (2000)

    Book  Google Scholar 

  13. Krishnapriyan, H.K.: On evaluating the characteristic polynomial through symmetric functions. J. Chem. Inf. Comput. Sci. 35, 196–198 (1995)

    Article  Google Scholar 

  14. Li, B., Shen, Z.: Ricci curvature tensor and non-Riemannian quantities. Canad. Math. Bull. 58(3), 530–537 (2015)

    Article  MathSciNet  Google Scholar 

  15. Li, B., Shen, Z.: Sprays of isotropic curvature. Internat. J. Math. 29(1), 1850003, 12 (2018)

    Article  MathSciNet  Google Scholar 

  16. Matveev, V.S., Topalov, P.I.: Trajectory equivalence and corresponding integrals. Regul. Chaotic Dyn. 3(2), 30–45 (1998)

    Article  MathSciNet  Google Scholar 

  17. Sarlet, W.: A recursive scheme of first integrals of the geodesic flow of a Finsler manifold. SIGMA Symmetry Integr. Geom. Methods Appl. 3, 24 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Springer, New York (2001)

    Book  Google Scholar 

  19. Shen, Z.: On some non-Riemannian quantities in Finsler geometry. Canad. Math. Bull. 56(1), 184–193 (2013)

    Article  MathSciNet  Google Scholar 

  20. Shen, Z.: On sprays with vanishing \(\chi \)-curvature. Int. J. Math. 32(10), 2150069 (2021)

    Article  MathSciNet  Google Scholar 

  21. Szilasi, J., Lovas, R.L., Kertész, D.C.: Connections. Sprays and Finsler Structures. World Scientific, Hackensack, NJ (2014)

    MATH  Google Scholar 

  22. Tabachnikov, S.: Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid. Comment. Math. Helv. 74, 306–321 (1999)

    Article  MathSciNet  Google Scholar 

  23. Youssef, N.L.: Semi-projective changes. Tensor, N.S. 55, 131–141 (1994)

Download references

Acknowledgements

This manuscript has no associated data.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Alexandru Ioan Cuza University, Iaşi, Romania

    Ioan Bucataru & Oana Constantinescu

  2. Department of Mathematics, Gheorghe Asachi Technical University, Iaşi, Romania

    Georgeta Creţu

Authors
  1. Ioan Bucataru
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oana Constantinescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Georgeta Creţu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioan Bucataru.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bucataru, I., Constantinescu, O. & Creţu, G. First integrals for Finsler metrics with vanishing \(\chi \)-curvature. Ann Glob Anal Geom 62, 815–827 (2022). https://doi.org/10.1007/s10455-022-09872-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-022-09872-y

Keywords

Mathematics Subject Classification

Search

Navigation